Question

Having survived the meteor impact, thanks to some last minute evasive maneuvers, the Mathonauts now set their sights on their Interstellar Headquarters. The Interstellar Headquarters orbits the Earth based on the equation y2 + x2 = 40,000. Using the original trajectory of the ship and complete sentences, explain to the pilot how to find where the ship’s path will cross the Interstellar Headquarters’s path. the original equation is y=6x+2

Answer 1

@Kfins99

Answer 2

you have two equations with y and x you have to use one and plug it into the other to solve the system.

since the second equation is in terms of y you should plug this into the other to solve for x

Answer 3

thank you @PhysicsGuru but how would the equation look

Answer 4

instead of
\[y^2 + x^2 = 40,000 \]
you will plug in 6x+2 for y

Answer 5

try to do some of the steps... you have to factor the term and solve the quadratic equation for x

Answer 6

yeah i tried to do that but it doesnt work cuz of the square i get stuck in the process @PhysicsGuru

Answer 7

ok so we plug in 6x +2 and get
\[(6x+2)^2 + x^2 = 40,000\]
which can be written as
\[(6x+2)(6x+2) + x^2 = 40,000\]

Answer 8

You have to use FOIL for the term on the left

Answer 9

but what about the x^2?

Answer 10

@PhysicsGuru

Answer 11

after everything is on the left side of the equation you have x^2 left

Answer 12

After you foil the term on the left you will add the x^2 afterwards ... always go from more complex to least complex when solving for a variable

Answer 13

tell me how you foil the term on the left

Answer 14

36x^2+12x+12x+4=40000 am i doing it right? @PhysicsGuru

Answer 15

Yes however you are missing the extra x^2 term from the left side of the equation

Answer 16

oh yah so 37^2? @PhysicsGuru

Answer 17

@satellite73 do you think you can help

Answer 18

@ns_69

Answer 19

@goformit100 can you pls helpppp

Answer 20

You have been helped by so many users. Sir How can I hep you ?

Answer 21

i gett lost in the process do you get the problem? i just need you to tell me if im doing it right or not.

Answer 22

@goformit100

Answer 23

. You can substitute your original trajectory (y=5x+1) into the equation that the Interstellar Headquarters is based on (y2+x2=40,000). You will be with the equation 2(5x+1)+x2=40,000 and you will solve for X. After solving for X, you will have X=3333.12 for your value. You will now take this value back into one of your original equations. You will now have the equation y=5(3333.17)+1 and will solve for Y. Your value for Y will be Y=16666.85. You now know the ship’s path will cross the Interstellar Headquarters’ at the coordinate (3333.17, 16666.85).

Answer 24

that doesnt work cuz its to the power and i saw this oh yahoo answers also

Answer 25

ohk that was from yahoo ans fine!

Answer 26

@phi can plzzzzzzz help

Answer 27

\[ y^2 + x^2 = 40,000 \\y=6x+2
\]
as already posted up above, replace y in the first equation with 6x+2
\[ (6x+2)^2 + x^2 = 40000 \]

can you multiply out (6x+2)(6x+2) ?
Did you try ?

Answer 28

yeah i got 37^2+24x+4=40,000

Answer 29

I assume you mean (putting in the x^2)
\[ 37x^2+24x+4=40,000 \]
subtract 40000 from both sides
\[ 37x^2 +24x -39996=0 \]
that is a bit ugly looking. But we can use the quadratic formula (with a calculator) to find 2 x values that will work.
then you can find the corresponding y values using y= 6x+2

Do you know how to use the quadratic formula?

Answer 30

x=-B+-SQRT b^2-4ac/2a you want me to solve it and tell you?

Answer 31

do it step by step. first what is b^2 - 4ac and the sqr of b^2 - 4ac

Answer 32

ok 24^2-4(37)(-3996)

Answer 33

and we need a number

Answer 34

ok that = -590832?

Answer 35

no, be careful of the signs. minus times minus is plus

Answer 36

ok so 590832?

Answer 37

is your calculator broken ?

Answer 38

lol sorry

Answer 39

wait is it 768.655

Answer 40

type
24^2-4(37)(-3996)=
into google. (But I would try to get the same answer using the calculator... you do want to be able to use a calculator... it is a useful thing to know)

Answer 41

I feel like your going to kill me if i dont get this one i swear i typed it as you said but the number is still weird here you go 591984

Answer 42

oops, it should be
24^2-4(37)(-39996)=

Answer 43

ok its 591984

Answer 44

and google says 5919984

Answer 45

that is missing a number. you should get 5,919,984

Answer 46

so thats x?

Answer 47

remember the last number is -39,996 NOT -3,996

Answer 48

so far you found b^2 -4ac = 5,919,984
you now need the square root of that number (and save it because you will need to use a few times)

Answer 49

the sqrt is 2433.101724

Answer 50

ok , now you know \( \sqrt{b^2 -4ac}= 2433.101724 \)
now let's find one of the x values
x= (-b + 2433.101724)/(2a)

Answer 51

lol ok jst hold up im going to solve this

Answer 52

ok i got 32.5554287

Answer 53

ok,now find y using y = 6x+2

Answer 54

197.3325722

Answer 55

yes, so you have one point where the line intersects the ellipse
(32.555, 197.333) (rounded to the nearest thousandth)

next, find the other x value, using the minus sqrt(b^2-4ac):
x= (-b - 2433.101724)/(2a)

Answer 56

other square you mean x^2 or jst x

Answer 57

(33 ,197)

Answer 58

you lost some minus signs

Answer 59

what minus signs

Answer 60

x= (-b - 2433.101724)/(2a)

that will give a negative x.
remember the quadratic equation gives two answers because we do ± sqr(b^2-4ac)
we are now doing
\[ x = (-b - \sqrt{b^2-4ac})/(2a) \]

Answer 61

oh yeah i totaly forgotot so itss (33,197) (-33,-197)

Answer 62

yes. I would round to about 3 decimals, just to show the answer is not an exact integer.
(unless the question says round to the nearest integer)

Answer 63

ok Thank you Soooo muchhhhh Phi! i have been stuck on this for quite a while!! :)

Answer 64

Here is a graph of your problem. (I used geogebra, which you can download for free)

Answer 65

Thank YOU!!!!

Answer 66

yw